Kloosterman sign changes with moduli having at most five prime factors

Abstract

On square-free moduli $q\in(X,2X]$ having at most five prime factors, we prove that each sign of the normalized Kloosterman sum $\operatorname{Kl}(1;q)$ occurs $\gg X/\log X$ times. This improves the recent unconditional result of Zhang and Zhong for moduli with at most six prime factors. Building on their analytic estimates and optimized Selberg sieve, we replace their truncated divisor penalty by a geometric half-weight. The new weight retains the $P_5$ exclusion threshold and is a positive linear combination of two standard two-parameter truncated divisor weights, so the Zhang--Zhong transference argument applies without alteration. After transference, the relevant pointwise coefficient is reduced from $5/16$ to $5/32$, which yields a positive final sieve margin.